Phonon dispersion relation: Difference between revisions
No edit summary |
|||
| Line 58: | Line 58: | ||
Equation 6 is an eigenfunction where <math>\xi</math> represents the eigenvalues, and <math>\vec{A}</math> represents the eigenvectors of the dynamical matrix <math>Q(\vec{k})</math>. We may notice each given <math>\vec{k}</math> coressponds to a unique <math>Q(\vec{k)}</math> and by solving the eigenvalues of <math>Q(\vec{k)}</math>, finally we can obtain the relation between <math>\vec{k}</math> and <math> \omega </math>, which is the '''phonon dispersion relation'''. |
Equation 6 is an eigenfunction where <math>\xi</math> represents the eigenvalues, and <math>\vec{A}</math> represents the eigenvectors of the dynamical matrix <math>Q(\vec{k})</math>. We may notice each given <math>\vec{k}</math> coressponds to a unique <math>Q(\vec{k)}</math> and by solving the eigenvalues of <math>Q(\vec{k)}</math>, finally we can obtain the relation between <math>\vec{k}</math> and <math> \omega </math>, which is the '''phonon dispersion relation'''. |
||
==TEST CASE== |
|||
[[media:Graphene_phonon.m.txt | Graphene_phonon.m]] [[media:Graphene_md.tcl.txt | Graphene_md.tcl]]. |
|||
Revision as of 06:13, 15 June 2012
Phonon Dispersion Relation
Yanming Wang, Saad Bhamla and Wei Cai
May, 2012
METHODS
Small displacement method for calculating the phonon dispersion relation
For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1
![{\displaystyle V(\{{\vec {r}}_{i}\})=V(\{{\vec {r}}_{i}\})+{\frac {1}{2}}\sum _{ij}{\vec {u}}_{i}^{T}\cdot [{\frac {d^{2}V}{d{\vec {r}}_{i}{\vec {r}}_{j}}}]\cdot {\vec {u}}_{j}\qquad (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81cc82825c7baf81190f8cb3b221ef6fc9564c0d)
where is the potential function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{ r}_{i}} is the coordinate of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ith} atom, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{ R}_{i}} is the position of the $ith$ atom at emuilibrium and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}} is a small displacement from the equilibrium position: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}_{i} = \vec{r}_{i} - \vec{R}_{i} } .
In Equation 1, the linear terms in the Taylor expansion are at 0K with the minimum energy state and the higher order terms are neglected using the Harmonic approximation. The second derivative of the potential energy evaluated at the equilibrium position and is called the force constant matrix or the Hessian matrix and is obtained from MD++ using the calHessian function.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_{ij} =[\frac{d^{2}V}{d \vec{r}_i \vec{r}_j}] = \left[ {\begin{array}{ccc} \frac{\partial^2V}{\partial x_i\partial x_j} & \frac{\partial^2V}{\partial x_i\partial y_j} & \frac{\partial^2V}{\partial x_i\partial z_j} \\ \frac{\partial^2V}{\partial y_i\partial x_j} & \frac{\partial^2V}{\partial y_i\partial y_j} & \frac{\partial^2V}{\partial y_i\partial z_j} \\ \frac{\partial^2V}{\partial z_i\partial x_j} & \frac{\partial^2V}{\partial z_i\partial y_j} & \frac{\partial^2V}{\partial z_i\partial z_j} \\ \end{array}} \right] \qquad(2) }
From Equation 2, we can calculate the force on the atoms, as shown in Equation 3 as follows, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{f}_{i} = -\frac{d V}{d \vec{r}_{i}}\bigg|_{\vec{R}_{i}} = - \sum_{j} K_{ij} \vec{u}_{j} \qquad(3) }
This force acting on the atoms leads to vibrations in the lattice. Applying the Newtonian equation of motion, we obtain Equation 4,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{f}_{i} = m\ddot{u} = - \sum_{j} K_{ij} \vec{u}_{j} \qquad(4) }
Obviously, equation 4 is a second order differential equation about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}} ,and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}} can be derived from a general solution of a plane wave shown in Equation 5,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}(\vec{R},t) = \vec{A}\cdot exp[i(\vec{k} \cdot \vec{R} - \omega t)] \qquad(5) }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{A}} determines the amplitude and direction of vibration, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} represents the vibration modes of the system, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{k}} represents the wave vector. Substituting Equation 5 in Equation 4, we obtain the following eigenfunction expression:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{A} \cdot \xi = Q(\vec{k)} \cdot \vec{A} \qquad(6) }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi = m \omega^{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(\vec{k)}} is the dynamical matrix defined by:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(\vec{k)} = \sum_{j} K_{ij} \cdot exp[i(\vec{k} \cdot \vec{R} - \omega t)] \qquad(7) }
Equation 6 is an eigenfunction where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi} represents the eigenvalues, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{A}} represents the eigenvectors of the dynamical matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(\vec{k})} . We may notice each given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{k}} coressponds to a unique Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(\vec{k)}} and by solving the eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(\vec{k)}} , finally we can obtain the relation between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{k}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega } , which is the phonon dispersion relation.